Okay, I'm gonna correct some things about... basically every character whose jumping can be questioned. In almost every case it is done in limited cases or they just haven't seen explicitly jumping while nothing really says that they CAN'T, just that they HAVEN'T.
: Can and has jumped. In Pikmin 2, to leave any cave both Olimar and either Louie or The President (depending on what point of the game you are at) must leap up into a Geyser before being launched out with their Pikmin following immediately after. While this is a cutscene, it's a proof that Olimar can and has canonically jumped... legitimately a height that's about two times that of his own.
: Jumps at the end of the animation intro for Pokemon Yellow. In Pokemon Stadium, Pikachus would jump in their victory animatio during the charging mini-game in BOTH versions of this game. He did jumps of joy and leapt off ledges in his spin-off game back on the N64, Hey You, Pikachu! Pokemon Snap I believe had one photo op of a Pikachu that jumped. That is if you want to omit the entire anime function of Pikachu or the idle animations from the Mystery Dungeon games.
: Jumped away from and at their opponent in the Pokemon Blue intro.
: Jumps in the victory in a minigame, in the intro, and during some attacks in Pokemon Stadium 2.
: Psychic. See Ness.
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: Jumping Mini-Game in Fire Red/Leaf Green if you needed more justification.
: Charizard used Fly! I don't know of a single organism with legs that can fly but not jump.
: I'm going to write this off from his film appearance because that is what his voice is based on.
: Jumps in a number of attack animations. For one example, Acrobatics.
: There was in fact an official anime for Marth's original game. He jumped in it.
: Had yet to appear in a game, so everything was canon in Melee.
: His game had actual jumping to get up and down ledges. Also Aether.
: Jumps in the cutscene fight with Chrom.
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: Jumps in their critical animation for swords/axes.
: Jumped off many a ledge. Jump Slash
: is
who jumped.
: Flew. See Charizard.
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: Every part of their moveset is made up of their theme.
: He doesn't actually jump, it's him manipulating his own body using PSI energy. That's why it has that Sci-Fi sound effect. (Not in any way canon but it's a good enough headcanon).
: Actually has jumped (at the end of the Thunder Tower to grab onto Rope Snake dangling from a helicopter). But also see Ness.
: Is a very cartoon-y dog can we just let this one pass.
: Jump rope in his training montage.
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: Jumps in many of their Party-style games. For gunner specifically, tie to the Metroid game mode in Nintendo Land.
: Is a diety. You can't really complain about a Goddess jumping when she canonically can teleport. Also, pretty sure she jumped in the Palutena's Revolting Dinner animation.
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: Jumps out of pitfall traps.
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: ...Can we just assume that fitness professionals that clearly have manipluation of their legs can do simple jumps?
The difference between these characters and Captain Toad is that
the crux of his gameplay is that he can't jump. Both in his missions in 3D World and his new game, the point is that Captain Toad can not jump and thus must navigate the maps in a much less direct way. This is like if Megaman couldn't shoot, Shulk didn't have the Monado, or if Sonic wasn't fast.